Simulations of estimating causal effects using undersmoothed HAL
Seraphina Shi
2023-03-17
Simulation 3
Data structure: \(O = (W, A, Z, Y)\)
- U - exogenous variables
- W - baseline covariate that is a measure of body condition
- A - treatment level based on W, continuous between 0 and 5
- Z - intermediate curve based on W and A
- Y - outcome, indicator of an event ?
Underlying data generating process, \(P_{U,X}\)
- Exogenous variables:
- \(U_A \sim Normal(\mu=0, \sigma^2 = 1^2)\)
- \(U_A \sim Normal(\mu=0, \sigma^2 = 2^2)\)
- \(U_Z \sim Uniform(min = 0, max = 1)\)
- \(U_Y \sim Uniform(min = 0, max = 1)\)
- Structural equations F and endogenous variables:
- \(W = U_W\)
- \(A = bound(2 - 0.5W + U_A, min=0, max=5)\)
- \(Z = \mathbf{I}[U_Z < expit(2-W-A)]\)
- \(Y = \mathbf{I}[U_Y < expit(-10 - 3W + 2A + Z(5 + 2 sin(A^2) - 20 \mathbf{I}(A>4)))]\)
## W A Z Y
## Min. :-3.71792 Min. :0.000 Min. :0.0000 Min. :0.0000
## 1st Qu.:-0.65715 1st Qu.:0.606 1st Qu.:0.0000 1st Qu.:0.0000
## Median : 0.01527 Median :1.987 Median :0.0000 Median :0.0000
## Mean : 0.01455 Mean :2.098 Mean :0.4828 Mean :0.2507
## 3rd Qu.: 0.68169 3rd Qu.:3.347 3rd Qu.:1.0000 3rd Qu.:1.0000
## Max. : 4.34792 Max. :5.000 Max. :1.0000 Max. :1.0000## Summary of A given W < -1:## Min. 1st Qu. Median Mean 3rd Qu. Max.
## 0.000 1.331 2.656 2.623 3.930 5.000## Summary of A given -1 < W <= 0:## Min. 1st Qu. Median Mean 3rd Qu. Max.
## 0.0000 0.9508 2.2621 2.3268 3.6176 5.0000## Summary of A given 0 < W <= 1:## Min. 1st Qu. Median Mean 3rd Qu. Max.
## 0.0000 0.3793 1.7606 1.9012 3.0529 5.0000## Summary of A given 1 < W:## Min. 1st Qu. Median Mean 3rd Qu. Max.
## 0.000 0.000 1.285 1.536 2.569 5.000
n = 500
CV HAL
results
## CV selected lambda (from one sample): 0.00179663135270864
1000 repetition
## The average of CV selected lambdas (from 1000 sample): 0.00167069905743632## z=1:
## z=0:
Globally Undersmoothed HAL
results
## Undersmoothed lambda: 0.000291732819942546
## which is 0.162377673918872 * lambda_CV
1000 repetition
## The average of unsersmoothed lambda (from 1000 sample): 0.000270475505996292
## which is 0.155768441258726 * the average of 1000 lambda_CV## z=1:
## z=0:
CV vs Undersmoothing
## z=1:
## z=0:
n = 1000
CV HAL
results
## CV selected lambda (from one sample): 0.000284242765357572
1000 repetition
## The average of CV selected lambdas (from 1000 sample): 0.00072537755501008## z=1:
## z=0:
Globally Undersmoothed HAL
results
## Undersmoothed lambda: 4.61546790670303e-05
## which is 0.162377673918872 * lambda_CV
1000 repetition
## The average of unsersmoothed lambda (from 1000 sample): 0.000103940431379792
## which is 0.146789904758241 * the average of 1000 lambda_CV## z=1:
## z=0:
CV vs Undersmoothing
## z=1:
## z=0: